31.6 problem 187

Internal problem ID [11010]
Internal file name [OUTPUT/10267_Sunday_December_31_2023_11_33_33_AM_40075043/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form \((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 187.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _with_linear_symmetries]]

Unable to solve or complete the solution.

\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{3}+a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y=0} \]

Maple trace Kovacic algorithm successful

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   Solution has integrals. Trying a special function solution free of integrals... 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Moebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   No special function solution was found. 
<- Kovacics algorithm successful`
 

✓ Solution by Maple

Time used: 0.454 (sec). Leaf size: 49

dsolve(x^3*diff(y(x),x$2)+(a*x^3-x^2+a*b*x+b)*diff(y(x),x)+a^2*b*x*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{-a x} \left (c_{2} \left (\int \frac {x \,{\mathrm e}^{\frac {2 a \,x^{3}+2 a b x +b}{2 x^{2}}}}{\left (a x +1\right )^{2}}d x \right )+c_{1} \right ) \left (a x +1\right ) \]

✓ Solution by Mathematica

Time used: 1.347 (sec). Leaf size: 70

DSolve[x^3*y''[x]+(a*x^3-x^2+a*b*x+b)*y'[x]+a^2*b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 e^{a K[1]+\frac {2 a K[1] b+b}{2 K[1]^2}} K[1]}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \]